Operator-theoretic infrared renormalization and construction of dressed 1-particle states in non-relativistic QED

TL;DR

This paper applies operator-theoretic infrared renormalization to a non-relativistic QED model, constructing dressed 1-particle states and analyzing their properties as the infrared cutoff vanishes.

Contribution

It develops new analytic tools to control the renormalization flow of the marginal interaction term in non-relativistic QED, leading to the construction of dressed states.

Findings

Existence of eigenvalues for all positive infrared cutoffs.

Eigenvectors converge to dressed states as cutoff vanishes.

Dressed states are not in Fock space in the zero cutoff limit for nonzero momentum.

Abstract

We consider the infrared problem in a model of a freely propagating, nonrelativistic charged particle of mass 1 in interaction with the quantized electromagnetic field. The hamiltonian of the system is regularized by an infrared cutoff $\ssig\ll 1$ and an ultraviolet cutoff $\Lambda\sim 1$ in the interaction term, in units of the mass of the charged particle. Due to translation invariance, it suffices to study $\Hps$, the restriction of the hamiltonian to the fibre Hilbert space of the conserved momentum operator associated to total momentum $p\in\R^3$. Under the condition that the coupling constant $g$ and the conserved momentum $p$ are sufficiently small, the following statements hold: (1) For every $\ssig>0$, $E_0:=inf spec \Hps$ is an eigenvalue with corresponding eigenvector $\Omega[p,\ssig]\in\Hp$. (2) For all $\ssig\geq0$, the first and second derivatives of $(\Egrd-\frac{|p|^2}{2})$ are $O(g^\delta)$ for some $\delta>0$. (3) $\Omega[p,\ssig]$ is not an element of the Fock space $\Hp$ in the limit $\ssig\to0$, if $|p|>0$. Our proofs are based on the operator-theoretic renormalization group of V. Bach, J. Fr\"ohlich, and I.M. Sigal. The key difficulty in the analysis of this system is connected to the strictly marginal nature of the leading interaction term, and a main issue in our exposition is to develop analytic tools to control its renormalization flow.